Modified minimum principal stress estimation formula based on Hoek–Brown criterion and equivalent Mohr–Coulomb strength parameters

The most critical parameter for determining equivalent values for the Mohr–Coulomb friction angle and cohesion from the nonlinear Hoek–Brown criterion is the upper limit of confining stress. For rock slopes, this value is the maximum value of the minimum principal stress (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma_{3,\max }^{\prime }$$\end{document}σ3,max′) on the potential failure surface. The existing problems in the existing research are analyzed and summarized. Using the finite element method (FEM), the location of potential failure surfaces for a wide range of slope geometries and rock mass properties are calculated using the strength reduction method, and a corresponding finite element elastic stress analysis was carried in order to determine \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma_{3,\max }^{\prime }$$\end{document}σ3,max′ of the failure surface. Through a systematic analysis of 425 different slopes, it is found that slope angle (β) and geological strength index (GSI) have the most significant influence on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma_{3,\max }^{\prime }$$\end{document}σ3,max′ while the influence of intact rock strength and the material constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{i}$$\end{document}mi are relatively small. According to the variation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma_{3,\max }^{\prime }$$\end{document}σ3,max′ with different factors, two new formulas for estimating \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma_{3,\max }^{\prime }$$\end{document}σ3,max′ are proposed. Finally, the proposed two equations were applied to 31 real case studies to illustrate the applicability and validity.

The most critical parameter for determining equivalent values for the Mohr-Coulomb friction angle and cohesion from the nonlinear Hoek-Brown criterion is the upper limit of confining stress. For rock slopes, this value is the maximum value of the minimum principal stress ( σ ′ 3,max ) on the potential failure surface. The existing problems in the existing research are analyzed and summarized. Using the finite element method (FEM), the location of potential failure surfaces for a wide range of slope geometries and rock mass properties are calculated using the strength reduction method, and a corresponding finite element elastic stress analysis was carried in order to determine σ ′ 3,max of the failure surface. Through a systematic analysis of 425 different slopes, it is found that slope angle (β) and geological strength index (GSI) have the most significant influence on σ ′ 3,max while the influence of intact rock strength and the material constant m i are relatively small. According to the variation of σ ′ 3,max with different factors, two new formulas for estimating σ ′ 3,max are proposed. Finally, the proposed two equations were applied to 31 real case studies to illustrate the applicability and validity.

Abbreviations σ 1
Major effective principal stress σ 3 Minor effective principal stress σ ci Unconfined compressive strength of intact rock σ cm Unconfined compressive strength of rock mass σ t Tensile strength of rock mass σ v Gravitational stress σ ′ 3n Normalized upper limit of confining stress σ ′

3,max
Upper limit of confining stress over the equivalent Mohr-Coulomb and Hoek-Brown criteria are considered σ a

3,max
Appropriate value of σ 3,max obtained from elastic stress analysis σ Hoek-Brown constant for rock mass m i Hoek-Brown constant for intact rock s Hoek-Brown constant for rock mass a Hoek-Brown constant for rock mass c′ Equivalent cohesion φ′ Equivalent friction angle GSI Geological strength index D Disturbance factor γ Unit weight H Slope height where m i is a material constant for intact rock, GSI is the geological strength index which depends on rock mass characterization and commonly varies from 0 to 100; D is a factor which depends upon the degree of disturbance due to blast damage and stress relaxation and varies from 0 to 1. The GSI classification system is based upon the assumption that the rock mass contains sufficient number of 'randomly' oriented discontinuities such that it behaves as an isotropic mass. Therefore, the control failure of a single discontinuous structure is beyond its range, which will lead to highly anisotropic mechanical behavior.
In line with the above discussion, it is important to realise the research in this paper will be subject to the same limitations that underpin the Hoek-Brown yield criterion itself.
Hoek 9 proposed a method to calculate the equivalent Mohr-Coulomb parameters based on instantaneous rock mass properties for: (1) a specified effective normal stress, (2) a specified minor principal effective stress, and (3) a condition in which the rock mass uniaxial compressive strength is the same for both the Hoek-Brown and Mohr-Coulomb criteria. In 1997, Hoek and Brown 8 revised the method of calculating the equivalent Mohr-Coulomb parameters according to the generalized Hoek-Brown criterion. It is recommended that the maximum value of the minimum effective principal stress generally be 0.25σ ci , and the estimated c value using this method be decreased by 25% to avoid overestimating the rock mass strength. Also, for rock slopes, the effective normal stress on the potential failure surface of the slope may be small, so the maximum value of the minimum effective principal stress 0.25σ ci should be applied cautiously, otherwise the rock mass shear strength mass may be overestimated. For rock slopes, a minor principal stress range of 0 < σ 3 < σ v can be used, where σ v = depth × unit weight of the rock mass 18 . In this case, depth is defined as the average depth of a failure surface in which a circular type can be assumed.
The equations for determining the equivalent cohesion and friction angle proposed by Hoek et al. in 2002 and2018 are 5,19 : with where γ is the rock mass unit weight, H is the slope height, and σ cm is the rock mass global strength, which is expressed as follows:  11 found that, for steep slopes (i.e. greater than 45°), the safety factors calculated using the equivalent friction angle and cohesive strength obtained from Eqs. (5-9) are significantly higher due to the deviation of the estimated σ ′ 3,max ; therefore, they suggested the following modified power functions to estimate σ ′ 3,max : where βis the slope angle.
Renani and Martin 13 also studied the estimation of σ ′ 3,max through systematic slope stability analysis and found that using the σ ′ 3,max calculated from Eq. (8) resulted in a 14% overestimation of the safety factor on average with higher discrepancies for steeper slopes, which more importantly led to drastic overestimation of the normalized failure area by an average of 79%. And what is more, they found that σ ′

3,max
γ H is almost independent of σ cm γ H and is primarily controlled by the slope angle; therefore, Renani and Martin 13 proposed the following equation to estimate σ ′ 3,max : Equation (12) is obtained from an analysis of slopes with a range of parameters (Table 1).β, m i , GSI, and D almost cover the whole range of possible values (Table 1). Only σ ci γ H covers a narrow range of possible values. For example, when slope height H = 100 m, and γ = 0.027 MN/m 3 , then σ ci = 0.27-27 MPa, this situation represents only a small portion of natural rock slopes, therefore when σ ci γ H > 10 , the applicability of Eq. (12) needs to be verified.
The magnitude of the minimum principal stress on the potential failure surface of the rock slope is primarily related to its development location, which is not only related to slope angle but also to the intact rock strength and rock mass integrity. However, Eq. (8) does not consider the influence of slope angle and Eq. (12) does not take into account the effect of the intact rock strength and rock mass integrity. Although all factors are considered in Eqs. (10) and (11), the slope angle is divided into two cases: less than 45°and greater than or equal to 45°, which fails to consider the effect of a continuously changing slope angle on the minimum principal stress magnitude. In view of these problems that still exist in the current research, this contribution aims to propose a new estimation formula for σ ′ 3,max on the potential failure surface of the slope by extending the range of σ ci γ H > 10 in Table 1.

Methodology
In order to establish the estimation formula of σ ′ 3,max on the slope potential failure surface, the finite element strength reduction method for generalized Hoek-Brown criterion was adopted to calculate the location of the potential failure surface for a wide range of slope geometries and rock mass properties. Then the corresponding finite element elastic stress analysis was carried out in order to determine the value of σ ′ 3,max on the failure surface (based on the method proposed by Renani and Martin in 13 ). The σ ′ 3,max values on 425 potential slope failure surfaces are calculated and used in a statistical analysis to obtain new estimation formulas.
It is noteworthy to highlight that in reality, even for the rock mass exhibits isotropic characteristics, the presence of distinct structural planes and fault can lead to deviations in the sliding surface of local slopes. However, these deviations are generally considered to be within acceptable limits.  www.nature.com/scientificreports/ Table 2 shows the range of slope parameters used in this study. Finite element strength reduction analysis and elastic stress analysis was carried out using RS2 software. The gravitational stress field had a horizontal to vertical in situ stress ratio of unity, rock mass deformation modulus E rm was estimated using the Eq. (13) 20 , the rock mass residual index is the same as the peak index, and Poisson's ratio was 0.28. Figure 1 shows the position and shape of the potential failure surface of the slope calculated using the finite element strength reduction method (corresponding to the maximum shear strain band), and Fig. 2 shows the minimum principal stress σ ′ 3,max on the potential failure surface.
The disturbance factor D was not considered in this study because the disturbed zone of the slope caused by blasting excavation is primarily limited to the shallow part of the slope; hence, D should not be used for the entire slope 19 . Due to the difference in blasting methods, slope shapes, and rock mechanical properties, the range of slope disturbance zones varies significantly. At present, there is no method to estimate this range, which makes considering the effect of D difficult.
The disturbance factor D is usually considered to take into account of the effects of reduction of GSI caused by construction disturbance. When the slope is analyzed for various GSI values, the individual effect of D can be safely ignored.

Impact analysis of various factors
As mentioned earlier, Eq. (8) does not consider the influence of slope angle, while Eq. (12) only considers the influence of slope angle without considering the intact rock strength and rock mass integrity. This study shows that for slopes with varying rock mass properties, there will be a large difference range of σ ′ 3,max when the range of σ ci γ H is between 10 and 50, even for the same slope angles. When the slope angle is 30°, the difference in σ ′

3,max
γ H can reach 0.5 (Fig. 3), indicating that in addition to the slope angle, the rock mass properties have a non-negligible impact on σ ′ 3,max .
σ ci to γH ratio. Homogeneous rock slope failure is related to the strength of the intact rock that composes the slope. Under the same conditions, the greater the uniaxial compressive strength of the rock block, the higher the slope stability and the deeper the potential failure surface. In this study, σ ci γ H is used to characterize the effect of rock block strength on σ ′ 3,max of the potential failure surface. Figure 4a- γ H increases slowly with increasing σ ci γ H , and the smaller the slope angle, the greater the increase (Fig. 4).

Geological Strength Index (GSI). The Geological Strength Index (GSI) reflects rock mass integrity and
is the most important factor affecting slope stability and the location of the potential failure surface. GSI has a significant impact on the magnitude of σ ′ 3,max on the potential failure surface. Figure 5a-e show the correlation between σ ′

3,max
γ H values increase exponentially with increasing GSI value at different slope angles, and the growth curves at different slope angles are basically parallel, indicating that the change in σ ′

3,max
γ H with GSI is not affected by the size of the slope angle (Fig. 5). Similarly, the change in σ ′

3,max
γ H with GSI is also not affected by σ ci γ H (Fig. 5a-e).
Material constant m i . m i is a material constant for the intact rock which depends upon the mineralogy, composition, and grain size of the intact rock 21 , which is obtained from laboratory. Hoek and Brown 19 proposed an approximate relationship between the compressive to tensile strength ratio, σ ci |σ t | , and the Hoek-Brown parameter m i : where |σ t | is the absolute value of the uniaxial tensile strength.
In order to examine the influence of m i on www.nature.com/scientificreports/ Slope angle. Slope angle not only affects the stress distribution in the rock mass but also affects the location of the potential failure surface, indicating that the effect of slope angle on σ ′ 3,max is significant. Numerous calculations show that when other conditions are the same, σ ′ 3,max decreases exponentially with increasing slope angle. Figure 7 shows the variation of

New equation for estimating appropriate range of confining stress
In "Impact analysis of various factors", the influence of various factors on σ ′ 3,max of the potential failure surface is analyzed. Results show that slope angle and GSI have the most significant influence on σ ′ 3,max , followed by the intact rock strength, while the m i has the least effect. According to the variation pattern of σ ′ 3,max with each factor, the following function is used as a fitting function:  Based on the above assumptions, an analysis was carried out on the calculation results of 425 slopes with a wide range of slope geometries and rock mass properties. Equation (15) was fit to these data, and the following best-fit equation was derived: The σ ′

3,max
γ H values predicted from Eq. (16) are plotted against appropriate values (finite element method results) in Fig. 8. The correlation between the predicted and appropriate σ ′

3,max
γ H values is reasonably close to the ideal 1:1 relationship of a perfect fit.

Validation using published data
In "New equation for estimating appropriate range of confining stress", two new equations for estimating appropriate range of confining stress have been established. To further validate the developed new equations, we have applied them to the various real cases of rock slopes 22 . The gathered data and calculation results are shown in Tables 3 and 4, respectively.
The root mean squared error (RMSE) is used as an indicator pf the misfit between the appropriate value and the predicted value. As can be seen in Fig. 12, the RMSE values form all modified predictive equations in this study are lower than the equations from previous studies.
When σ ci γ H < 10 , the predicted σ 3,max γ H given in Eqs. (16) and (17)  γ H given in Eqs. (10) and (11) are quite above the appropriate values with a mean absolute error of 0.1944. The prediction results of Eq. (12) are close to those of Eq. (16), but the accuracy is lower than that of Eq. (16) with RMSE of 0.0917 (Figs. 12, 13).
When σ ci γ H ≥ 10 , the predicted σ 3,max γ H given in Eqs. (16) and (17) are both very approach to the appropriate values. Meanwhile, the RMSE of the predicted value by Eqs. (16) and (17) are 0.035 and 0.0387, respectively. The predicted values of Eqs. (10) and (11) are higher than the appropriate values with RMSE being 0.1106. The prediction result of Eq. (12) is significantly smaller than the appropriate value with RMSE being 0.1686. (Figs. 12, 14).

Conclusions
When the Mohr-Coulomb criterion is used to analyze slope stability, it is commonly necessary to use the Hoek-Brown criterion to obtain equivalent Mohr-Coulomb shear strength parameters, where the maximum value of the minimum principal stress on the potential failure surface of the slope is the most important parameter. Based on the finite element strength reduction method and elastic analysis, this paper systematically analyzes 425 different slopes. Slope angle and GSI have the most significant influence on σ ′ 3,max , followed by intact rock strength and m i . σ ′ 3,max decreases with increasing slope angle and m i as a power function and increases with increasing GSI and σ ci /γ H as an exponential and power function, respectively. On this basis, two new formulas for estimating σ ′ 3,max are fit to the data, and the average errors of Eqs. (16) and (17)

Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.